On a Conjecture of Birch and Swinnerton-Dyer

[摘要] Let (E/mathbb{Q}) be an elliptic curve defined by the equation(y^2=x^3 +ax +b). For a prime (p, linebreak p midDelta=-16(4a^3+27b^2)eq 0), define [ N_p = p+1 -a_p =|E(mathbb{F}_p)|. ] As a precursor to their celebrated conjecture,Birch and Swinnerton-Dyer originally conjectured that for someconstant $c$, [ prod_{p leq x, p midDelta } frac{N_p}{p} sim c(log x)^r, quad x o infty. ] Let (alpha _p) and (eta_p) be the eigenvalues of the Frobenius at (p). Define [ilde{c}_n = egin{cases} frac{alpha_p^k + eta_p^k}{k}& n =p^k,p extrm{ is a prime, $k$ is a natural number, $pmid Delta$} .\ 0 & ext{otherwise}. end{cases}. ] and (ilde{C}(x)=sum_{nleq x} ilde{c}_n). In this paper, we establish theequivalence between the conjecture and the condition(ilde{C}(x)=mathbf{o}(x)). The asymptotic condition is indeedmuch deeper than what we know so far or what we can know under theanalogue of the Riemann hypothesis. In addition, we provide anoscillation theorem and an (Omega) theorem which relate to theconstant $c$ in the conjecture.

[学科分类] 数学（综合）